# Schwarz Lemma application problem

Schwarz lemma says that "Let $${\displaystyle \mathbf {D} =\{z:|z|<1\}}$$ be the open unit disk in the complex plane $${\displaystyle \mathbb {C} }$$ centered at the origin, and let $${\displaystyle f:\mathbf {D} \rightarrow \mathbb {C} }$$ be a holomorphic map such that $${\displaystyle f(0)=0}$$ and $${\displaystyle |f(z)|\leq 1}$$ on $${\displaystyle \mathbf {D} }$$. Then $${\displaystyle |f(z)|\leq |z|}$$ for all $${\displaystyle z\in \mathbf {D} }$$, and $${\displaystyle |f'(0)|\leq 1}$$. Moreover, if $${\displaystyle |f(z)|=|z|}$$ for some non-zero $${\displaystyle z}$$ or $${\displaystyle |f'(0)|=1}$$, then $${\displaystyle f(z)=az}$$ for some $${\displaystyle a\in \mathbb {C} }$$ with $${\displaystyle |a|=1}$$."

I am trying to use this to find an analytic function $$f: \mathbb{D} \rightarrow \mathbb{D}$$ such that $$f(1/8)=4/7$$ and $$f(4/5)=3/7$$? Could someone please explain? Thanks!

• How could you possibly use the Schwarz lemma to find a function with certain properties? the lemma doesn't say anything exists... Commented Apr 25, 2021 at 17:38
• As far as I know it does (last sentence of the first paragraph), may be I have misunderstood what you are saying ? @DavidC.Ullrich Commented Apr 25, 2021 at 18:17
• Ok, it's not true to say it doesn't show anything exists. It doesn't assert the existence of any holomorphic functions. Commented Apr 25, 2021 at 18:20

You cannot apply the Schwarz lemma directly because $$f(0) = 0$$ is not given.
But you can use its variant, the Schwarz-Pick theorem: $$\left|{\frac {f(z_{1})-f(z_{2})}{1-\overline {f(z_{1})}f(z_{2})}}\right|\leq \left|{\frac {z_{1}-z_{2}}{1-\overline {z_{1}}z_{2}}}\right|$$ to show that such a function does not exist.
• @Martin Thanks for your comment. I am thinking of constructing a map $g: \mathbb{D} \rightarrow \mathbb{D}$ that fixes the origin and then use Schwarz lemma. But I don't know how to construct $g$. Any idea? Commented Apr 25, 2021 at 18:13
• @GeetThakur: $g = T_1 \circ f \circ T_2$ where $T_1, T_2$ are suitable automorphisms of the unit disk. Commented Apr 25, 2021 at 18:17